Local Dynamics
The qualitative properties of dynamical systems do not change under a smooth change of coordinates (this is sometimes taken as a definition of qualitative): a singular point of the vector field (a point where v(x) = 0) will remain a singular point under smooth transformations; a periodic orbit is a loop in phase space and smooth deformations of the phase space cannot alter it being a loop. It is in the neighborhood of singular points and periodic orbits that the structure of a phase space of a dynamical system can be well understood. In the qualitative study of dynamical systems, the approach is to show that there is a change of coordinates (usually unspecified, but computable) that makes the dynamical system as simple as possible.
Read more about this topic: Dynamical System
Famous quotes containing the words local and/or dynamics:
“These native villages are as unchanging as the woman in one of their stories. When she was called before a local justice he asked her age. “I have 45 years.” “But,” said the justice, “you were forty-five when you appeared before me two years ago.” “Señor Judge,” she replied proudly, drawing herself to her full height, “I am not of those who are one thing today and another tomorrow!””
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